تسجيل الدخول

تسجيل

Modules-Multilinear Maps and Tensor Products

المؤلف:  David R. Wilkins

المصدر:  Algebraic Topology

الجزء والصفحة:  96-97

4-7-2017

2002

Let M₁, M₂, . . . , M_n be modules over a unital commutative ring R, and let P be an R-module. A function f: M₁ × M₂ × · · · × M_n → P is said to be R-multilinear if

              f(x₁, . . . , x_k−1, x`k + x``_k, x_k+1, . . . , x_n)

                            = f(x₁, . . . , x_k−1, x`k, x_k+1, . . . , x_n)

                                         + f(x₁, . . . , x_k−1, x``k, x_k+1, . . . , x_n)

and

               f(x₁, . . . , x_k−1, rx_k, x_k+1, . . . , x_n) = rf(x₁, . . . , x_k−1, x_k, x_k+1, . . . , x_n)

for k = 1, 2, . . . , n, for all x_l, x`_l, x``_l ∈ Ml (l = 1, 2, . . . , n), and for all r ∈ R.

(When k = 1 the list x₁, . . . , x_k−1 should be interpreted as the empty list in the formulae above; when k = n the list x_k+1, . . . , x_n should be interpreted as the empty list.) One can construct a module M₁ ⊗_R M₂ ⊗_R · · · ⊗_R M_n,  referred to as the tensor product of the modules M₁, M₂, . . . , M_n over the ring R, and an R-multilinear mapping

j_{M1×M2×···×Mn}: M₁ × M₂ × · · · × M_n → M₁ ⊗_R M₂ ⊗_R · · · ⊗_R M_n

where the tensor product and multilinear mapping j_{M1×M2×···×Mn} satisfy the following universal property:

given any R-module P, and given any R-multilinear function f: M₁ × M₂ × · · · × M_n → P, there exists a unique R-module homomorphism θ: M₁ ⊗_R M₂ ⊗_R · · · ⊗_R M_n → P such that f = θ ◦ j_{M1×M2×···×Mn}

This tensor product is defined to be the quotient of the free module F_R(M₁×M₂×· · ·×M_n) by the submodule K generated by elements of the free module that are of the form

i_{M1×M2×···×Mn} (x₁, . . . , x_k−1, x`_k + x``_k, x_k+1, . . . , x_n)

             − i_{M1×M2×···×Mn} (x₁, . . . , x_k−1, x`_k, x_k+1, . . . , x_n)

            − i_{M1×M2×···×Mn} (x₁, . . . , x_k−1, x``_k, x_k+1, . . . , x_n),

or are of the form

               i_{M1×M2×···×Mn} (x₁, . . . , x_k−1, rx_k, x_k+1, . . . , x_n)

                       − ri_{M1×M2×···×Mn} (x₁, . . . , x_k−1, x_k, x_k+1, . . . , x_n),

where x_l, x`_l, x``_l ∈ M_l  for l = 1, 2, . . . , n, and r ∈ R. There is an R-multilinear function

j_{M1×M2×···×Mn}: M₁ × M₂ × · · · × M_n → M₁ ⊗_R M₂ ⊗_R · · · ⊗_R M_n,  where j_{M1×M2×···×Mn}

is the composition π ◦ i_{M1×M2×···×Mn} of the natural embedding

i_{M1×M2×···×Mn}: M₁ × M₂ × · · · × M_n → F_R(M₁ × M₂ × · · · × M_n)

and the quotient homomorphism

            π: F_R(M₁ × M₂ × · · · × M_n) → M₁ ⊗_R M₂ ⊗R · · · ⊗_R M_n.